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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|364}}
{{EDO intro|152}}


== Theory ==
== Theory ==
364edo is [[consistent]] through the [[21-odd-limit]]. The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| -65 0 28 }} ([[oquatonic comma]]) in the [[5-limit]]; 65625/65536 ([[horwell comma]]), 390625/388962 ([[dimcomp comma]]), and 420175/419904 (wizma) in the [[7-limit]] (supporting [[fifthplus]] and [[oquatonic]]); 1375/1372, [[6250/6237]], [[19712/19683]], and [[41503/41472]] in the [[11-limit]] (as well as [[9801/9800]]); [[625/624]], [[1716/1715]], [[2080/2079]], [[2200/2197]], and 14641/14625 in the [[13-limit]] (as well as [[4096/4095]], [[4225/4224]], and [[10985/10976]]); [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 8624/8619 in the [[17-limit]] (as well as [[2431/2430]], [[4914/4913]], and [[5832/5831]]); [[1216/1215]], 1331/1330, 1540/1539, and [[1729/1728]] in the [[19-limit]].
152edo is a strong [[11-limit]] system, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] slightly sharp. It [[tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 32 -7 -9 }} ([[escapade comma]]) in the 5-limit; [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[16875/16807]] in the 7-limit; [[540/539]], [[1375/1372]], [[3025/3024]], [[4000/3993]], [[5632/5625]] and [[9801/9800]] in the 11-limit. It provides the [[optimal patent val]] for the 11-limit linear temperaments [[amity]], [[grendel]], and [[kwai]], and the 11-limit planar temperament [[laka]].
 
It has two reasonable mappings for 13, with the 152f val scoring much better. The 152f val tempers out [[352/351]], [[625/624]], [[640/637]], [[729/728]], [[847/845]], [[1188/1183]], [[1575/1573]], [[1716/1715]] and [[2080/2079]], [[support]]ing and giving an excellent tuning for amity, kwai, and laka. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit. The [[patent val]] tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1001/1000]], [[1573/1568]], and [[4096/4095]], providing the optimal patent val for the 13-limit rank-5 temperament tempering out 169/168, as well as some further temperaments thereof, such as [[octopus]].
 
[[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|364|columns=11}}
{{Harmonics in equal|152}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 364 factors into {{factorization|364}}, 364edo has subset edos {{EDOs| 2, 4, 7, 13, 14, 26, 28, 52, 91, 182 }}.
Since 152 factors into {{factorisation}}, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}.  
 
=== Miscellaneous properties ===
364edo can act as "pseudo-24024edo" in a sense that it can replicate being a multiple of [[11edo]], [[12edo]], [[13edo]] and [[14edo]]. It has 13 and 14 as its divisors, while at the same time supporting the Supermajor[11] scale from 91edo, which is a very precise temperament, and WorldCalendar[12] scale, which mimics 12edo. While it does not exactly replicate 11edo and 12edo, it comes close enough in harmonic parameters these edos are sought after.


== Regular temperament properties ==
== Regular temperament properties ==
Line 27: Line 28:
|-
|-
| 2.3
| 2.3
| {{monzo| 577 -364 }}
| {{monzo| 241 -152 }}
| {{mapping| 364 577 }}
| {{mapping| 152 241 }}
| −0.0766
| −0.213
| 0.0766
| 0.213
| 2.32
| 2.70
|-
|-
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| -65 0 28 }}
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| {{mapping| 364 577 845 }}
| {{mapping| 152 241 353 }}
| +0.0350
| −0.218
| 0.1698
| 0.174
| 5.15
| 2.21
|-
|-
| 2.3.5.7
| 2.3.5.7
| 65625/65536, 390625/388962, 420125/419904
| 4375/4374, 5120/5103, 16875/16807
| {{mapping| 364 577 845 1022 }}
| {{mapping| 152 241 353 427 }}
| −0.0098
| −0.362
| 0.1662
| 0.291
| 5.04
| 3.69
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 1375/1372, 6250/6237, 19712/19683, 41503/41472
| 540/539, 1375/1372, 4000/3993, 5120/5103
| {{mapping| 364 577 845 1022 1259 }}
| {{mapping| 152 241 353 427 526 }}
| +0.0366
| −0.365
| 0.1753
| 0.260
| 5.32
| 3.30
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625
| 352/351, 540/539, 625/624, 729/728, 1575/1573
| {{mapping| 364 577 845 1022 1259 1347 }}
| {{mapping| 152 241 353 427 526 563 }} (152f)
| +0.0245
| −0.494
| 0.1622
| 0.373
| 4.92
| 4.73
|-
| 2.3.5.7.11.13.17
| 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197
| {{mapping| 364 577 845 1022 1259 1347 1488 }}
| +0.0022
| 0.1599
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 625/624, 715/714, 1089/1088, 1216/1215, 1225/1224, 1331/1330, 1729/1728
| {{mapping| 364 577 845 1022 1259 1347 1488 1546 }}
| +0.0257
| 0.1620
| 4.91
|}
|}
* 152et (152fg val) has lower absolute errors in the 11-, 19-, and 23-limit than any previous equal temperaments. In the 11-limit it is the first to beat [[130edo|130]] and is superseded by [[224edo|224]]. In the 19- and 23-limit it is the first to beat [[140edo|140]] and is superseded by [[159edo|159]].
* It is best at the no-17 19- and 23-limit, in which it has lower relative errors than any previous equal temperaments. Not until [[270edo|270]] do we find a better equal temperament that does better in either of those subgroups.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 87: Line 76:
|-
|-
| 1
| 1
| 103\364
| 7\152
| 339.56
| 55.26
| 33/32
| [[Escapade]] / [[alphaquarter]]
|-
| 1
| 31\152
| 244.74
| 15/13
| [[Subsemifourth]]
|-
| 1
| 39\152
| 307.89
| 3200/2673
| [[Familia]]
|-
| 1
| 43\152
| 339.47
| 243/200
| 243/200
| [[Amity]] / [[paramity]]
| [[Amity]]
|-
|-
| 1
| 1
| 125\364
| 49\152
| 412.09
| 386.84
| 80/63
| 5/4
| [[Witch]]
| [[Grendel]]
|-
|-
| 1
| 1
| 149\364
| 63\152
| 491.21
| 497.37
| 3645/2744
| 4/3
| [[Fifthplus]]
| [[Kwai]]
|-
|-
| 1
| 1
| 151\364
| 71\152
| 497.80
| 560.53
| 4/3
| 242/175
| [[Gary]]
| [[Whoops]]
|-
| 2
| 7\152
| 55.26
| 33/32
| [[Septisuperfourth]]
|-
| 2
| 9\152
| 71.05
| 25/24
| [[Vishnu]] / [[acyuta]] (152f) / [[ananta]] (152)
|-
| 2
| 43\152<br />(33\152)
| 339.47<br />(260.53)
| 243/200<br />(64/55)
| [[Hemiamity]]
|-
|-
| 2
| 2
| 57\364
| 55\152<br />(21\152)
| 187.91
| 434.21<br />(165.79)
| 49/44
| 9/7<br />(11/10)
| [[Semiwitch]]
| [[Supers]]
|-
|-
| 4
| 4
| 30\364
| 63\152<br />(13\152)
| 98.90
| 497.37<br />(102.63)
| 18/17
| 4/3<br />(35/33)
| [[World calendar]]
| [[Undim]] / [[unlit]]
|-
|-
| 13
| 8
| 151\364<br />(11\364)
| 63\152<br />(6\152)
| 497.80<br />(36.26)
| 497.37<br />(47.37)
| 4/3<br />(?)
| 4/3<br />(36/35)
| [[Aluminium]]
| [[Twilight]]
|-
|-
| 26
| 8
| 151\364<br />(11\364)
| 74\152<br />(2\152)
| 497.80<br />(36.26)
| 584.21<br />(15.79)
| 4/3<br />(?)
| 7/5<br />(126/125)
| [[Iron]]
| [[Octoid]] (152f) / [[octopus]] (152)
|-
|-
| 28
| 19
| 151\364<br />(5\364)
| 63\152<br />(1\152)
| 497.80<br />(16.48)
| 497.37<br />(7.89)
| 4/3<br />(105/104)
| 4/3<br />(225/224)
| [[Oquatonic]]
| [[Enneadecal]]
|-
|-
| 91
| 38
| 151\364<br />(3\364)
| 63\152<br />(1\152)
| 497.80<br />(3.30)
| 497.37<br />(7.89)
| 4/3<br />(176/175)
| 4/3<br />(225/224)
| [[Protactinium]]
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[birdshite stalactite]]
* "athlete's feet" from ''razorblade tiddlywinks'' (2023) – [https://open.spotify.com/track/32c34U3syZDMAJkBzgh2pd Spotify] | [https://birdshitestalactite.bandcamp.com/track/athletes-feet Bandcamp] | [https://www.youtube.com/watch?v=lXqVaVn3SrA YouTube]


== Scales ==
[[Category:Amity]]
* WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30
[[Category:Grendel]]
[[Category:Kwai]]
[[Category:Laka]]
[[Category:Listen]]

Revision as of 14:44, 16 January 2025

← 363edo 364edo 365edo →
Prime factorization 22 × 7 × 13
Step size 3.2967 ¢ 
Fifth 213\364 (702.198 ¢)
Semitones (A1:m2) 35:27 (115.4 ¢ : 89.01 ¢)
Consistency limit 21
Distinct consistency limit 21

Template:EDO intro

Theory

152edo is a strong 11-limit system, with the 3, 5, 7, and 11 slightly sharp. It tempers out 1600000/1594323 (amity comma) and [32 -7 -9⟩ (escapade comma) in the 5-limit; 4375/4374, 5120/5103, 6144/6125 and 16875/16807 in the 7-limit; 540/539, 1375/1372, 3025/3024, 4000/3993, 5632/5625 and 9801/9800 in the 11-limit. It provides the optimal patent val for the 11-limit linear temperaments amity, grendel, and kwai, and the 11-limit planar temperament laka.

It has two reasonable mappings for 13, with the 152f val scoring much better. The 152f val tempers out 352/351, 625/624, 640/637, 729/728, 847/845, 1188/1183, 1575/1573, 1716/1715 and 2080/2079, supporting and giving an excellent tuning for amity, kwai, and laka. The optimal tuning of this temperament is consistent in the 15-integer-limit. The patent val tempers out 169/168, 325/324, 351/350, 364/363, 1001/1000, 1573/1568, and 4096/4095, providing the optimal patent val for the 13-limit rank-5 temperament tempering out 169/168, as well as some further temperaments thereof, such as octopus.

Paul Erlich has suggested that 152edo could be considered a sort of universal tuning.

Prime harmonics

Approximation of prime harmonics in 152edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.68 +0.53 +2.23 +1.31 -3.69 -2.32 +2.49 +3.30 -3.26 -0.30
Relative (%) +0.0 +8.6 +6.7 +28.2 +16.6 -46.7 -29.4 +31.5 +41.9 -41.3 -3.8
Steps
(reduced) 		152
(0) 		241
(89) 		353
(49) 		427
(123) 		526
(70) 		562
(106) 		621
(13) 		646
(38) 		688
(80) 		738
(130) 		753
(145)

Subsets and supersets

Since 152 factors into Lua error in Module:Utils at line 209: attempt to compare nil with number., 152edo has subset edos 2, 4, 8, 19, 38, 76.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [241 -152⟩ [⟨152 241]] −0.213 0.213 2.70
2.3.5 1600000/1594323, [32 -7 -9⟩ [⟨152 241 353]] −0.218 0.174 2.21
2.3.5.7 4375/4374, 5120/5103, 16875/16807 [⟨152 241 353 427]] −0.362 0.291 3.69
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 5120/5103 [⟨152 241 353 427 526]] −0.365 0.260 3.30
2.3.5.7.11.13 352/351, 540/539, 625/624, 729/728, 1575/1573 [⟨152 241 353 427 526 563]] (152f) −0.494 0.373 4.73
  • 152et (152fg val) has lower absolute errors in the 11-, 19-, and 23-limit than any previous equal temperaments. In the 11-limit it is the first to beat 130 and is superseded by 224. In the 19- and 23-limit it is the first to beat 140 and is superseded by 159.
  • It is best at the no-17 19- and 23-limit, in which it has lower relative errors than any previous equal temperaments. Not until 270 do we find a better equal temperament that does better in either of those subgroups.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 7\152 55.26 33/32 Escapade / alphaquarter
1 31\152 244.74 15/13 Subsemifourth
1 39\152 307.89 3200/2673 Familia
1 43\152 339.47 243/200 Amity
1 49\152 386.84 5/4 Grendel
1 63\152 497.37 4/3 Kwai
1 71\152 560.53 242/175 Whoops
2 7\152 55.26 33/32 Septisuperfourth
2 9\152 71.05 25/24 Vishnu / acyuta (152f) / ananta (152)
2 43\152
(33\152) 		339.47
(260.53) 		243/200
(64/55) 		Hemiamity
2 55\152
(21\152) 		434.21
(165.79) 		9/7
(11/10) 		Supers
4 63\152
(13\152) 		497.37
(102.63) 		4/3
(35/33) 		Undim / unlit
8 63\152
(6\152) 		497.37
(47.37) 		4/3
(36/35) 		Twilight
8 74\152
(2\152) 		584.21
(15.79) 		7/5
(126/125) 		Octoid (152f) / octopus (152)
19 63\152
(1\152) 		497.37
(7.89) 		4/3
(225/224) 		Enneadecal
38 63\152
(1\152) 		497.37
(7.89) 		4/3
(225/224) 		Hemienneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

birdshite stalactite
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